Diversification Penalties

Overview

Main Idea of the paper

Stochastic Dominance

Pareto Distributions and fat tails

Real world examples of fat tails

Conclusions of the paper

Considerations for our work

Main Idea

Specific - There are instances where a concentration of risk is preferable to a diversified portfolio. For example, when losses are Pareto distributed.

General - We make assumptions, shortcuts, heuristics all the time. That’s fine, but we need to be aware when our assumptions break down.

Stochastic Dominance

\[ X \le_{st} Y \leftrightarrow P(X \le x) \ge P(Y \le x) \forall x \in \mathbb{R} \]

A partial order of random variables

“\(X\) is smaller than \(Y\) in terms of first-order stochastic dominance”

Stochastic Dominance

Stochastic Dominance

The relation \(X \le_{st} Y\) means that all decision makers with an increasing utility function will prefer the loss \(X\) to the loss \(Y\)

(page 2)

Pareto Distribution

\[ CDF: P_{\alpha,\theta} = 1 - \left(\frac{\theta}{x}\right)^\alpha, x \ge \theta\\ Mean = \begin{cases} \infty & \alpha\le 1 \\ \frac{\alpha\theta}{\alpha - 1} & \alpha > 1 \end{cases}\\ \]

\[ \alpha \leftarrow \text{$``$shape''}\\ \theta \leftarrow \text{$``$scale''} \]

Pareto Distribution

Infinite mean on the distribution itself, not on (a finite set of) samples from the distribution.

Pareto Distribution

Examples of Pareto Risks

Earthquakes \(\alpha \in [0.5, 1.5]\)

Wind (some) \(\alpha \approx 0.7\)

Nuclear power accidents \(\alpha \in [0.6, 0.7]\)

Operational losses in banking \(\alpha \in [0.7, 1.2]\)

Cyber losses \(\alpha \in [0.6, 0.7]\)

Swiss Solvency Test (airline losses) \(\alpha = 1\)

Commercial Property losses at 2 Lloyds Syndicates \(\alpha\) “considerably less than 1”

Number of deaths in earthquakes and pandemics

(page 5)

Key Conclusions of the Paper

For \(X, X_1, ..., X_n \sim \text{Pareto}(\alpha), \alpha \in (0,1]\)

For \(\theta_1, ..., \theta_n \in \Delta_n\),

\[ X \le_{st}\sum_{i = 1}^n \theta_i X_i \]

And for \(t > 1\),

\[ P\left(\sum_{i = 1}^n \theta_i X_i > t \right) > P(X > t) \]

if \(\theta_i > 0\) for at least 2 \(i \in [n]\)

(Theorem 1, page 6)

Key Conclusions of the Paper

Having shares in multiple Pareto risks is less preferable to having full exposure to a single Pareto Risk

(This is not the case if the underlying distributions have finite mean)

Key Conclusions of the Paper

Generalised to random weights of random counts of Pareto Risks

It is less risky to insure one large policy than to insure any independent policies of the same type of ultra-heavy-tailed Pareto losses and thus the basic principle of insurance does not apply to ultra-heavy-tailed Pareto losses

(page 9)

Key Conclusions of the Paper

Also shows that under pooling of risk (e.g. sharing CAT losses):

Each insurer expects to suffer lower losses on their loss,

but each insurer will have a higher frequency of bearing a loss.

The combination leads to a higher probability of default of the insurer at any capital reserve level

(page 12)

Key Conclusions of the Paper

An insurance market cannot exist among a set of insurers each exposed to their own Pareto loss.

But under certain circumstances, including external entities with sufficient risk appetite can produce a market. Reinsurance still works.

My Conclusions